Visual Group Theory, Lecture 1.6: The formal definition of a group

Visual Group Theory, Lecture 1.6: The formal definition of a group

At last, after five lectures of building up our intuition of groups and numerous examples, we are ready to present the formal definition of a group. We conclude by proving several basic properties that are not built into the definition.

[Supremebubble]

let e and e' be identity elements. then e = ee' = e'. you don't even need associativity for this which is cool

[scitwi]

Can you show an example of an object which is _not_ associative?
What visualization techniques could we use to depict the operations on that object similarly to Cayley graphs?
My guess about the non-associativity is that there must be some operation which "seals" the combined object or "fuses" them together irreversibly, which is reflected by putting those parentheses around the operation. Then a*(b*c) is not the same as (a*b)*c, because in the first case, we first combine `b` and `c` and "seal" them, and then attach `a` to the fused result, but in the second case we first combine `a` and `b` and "seal" together, and then attach `c` to the fused result. This "sealing" is irreversible, so I guess a one-directional arrow is needed in the diagram, and with a step which cannot be reached in any other way than by that arrow (a "stepping stone" between two other nodes). Am I right?

[henlofrens]

on point 4, do the inverses also have to be in G? It makes sense that they are, but it's not explicitly stated.